If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:5:36

Sign of average rate of change of polynomials

CCSS.Math:

Video transcript

so we are given this function H of X and were asked over which interval does H have a positive average rate of change so like always pause this video and have a go at it before we do this together all right now let's work through this together and to start let's just remind ourselves what an average rate of change even is you can view it as the change in your value of the function for a given change in the underlying variable for given change in X we could also view this as if we want to figure out the interval we could say our X final minus our X initial and in the numerator it would be the value of our function at the X final minus the value of the function at our X initial now they aren't asking us to calculate this for all of these different intervals they're just asking us whether it is positive and if you look over here as long as our X final is greater than X initial in order to have a positive average rate of change we just need to figure out whether H at X final is greater than H at X initial if the value of the function at the higher end point is larger then the value of the function at the lower endpoint then we have a positive average rate of change so let's see if that's happening for any of these choices so let's see H of 0 this endpoint is going to be equal to 0 if I just say 1 8 times 0 minus 0 and H of 2 is equal to 1/8 times 2 to the third power is 8 so 1 8 times 8 is 1 minus 4 so that's going to be this is negative 3 and so we don't have a situation where H at our endpoint at our higher endpoint is actually larger this is a negative average rate of change so I'll rule this one out and actually just to help us visualize this I did go to desmos and graph this function and we can visually see that we have negative average rate of change from x equals zero to x equals two at x equals zero this is where our function is at x equals two this is where our function is and so you can see at x equals two our function has a lower value you could also think of the average rate of change as the slope of the line that connects the two end points on the function and so you can see it has a negative slope so we have a negative average rate of change between those two points now what about between these two so H of 0 we already calculated it as 0 and what is H of 8 well let's see that's 1/8 times 8 to the third power well if I if I do eight to the third power but then divide by eight that's the same thing as 8 to the second power so it's going to be 64 minus eight to the second power minus 64 so that's equal to zero so here we have a zero average rate of change because this numerator is going to be zero so we can rule that out and you see it right over here when X is equal zero our function is there when X is equal to eight our function is there and you can see that the slope of the line that connects those two points is zero so you have zero average rate of change between those two points now what about choice C so let's see H of 6 is going to be equal to 1/8 times 6 to the third power so let's see 36 times 6 is 180 plus 36 so that is going to be 2 16 to 16 minus 36 to 16 is 6 times 6 times 6 and then if we divide that by 8 that is going to be the same thing as this is 36 and then we have six eighths of 36 so this is going to be this is going to simplify to 3/4 times 36 minus 36 which is going to be equal to negative 9 you could have done it with a calculator or done some long division but hopefully what I just did makes some sense it's a little bit of arithmetic and so H of 6 we have our function is negative 9 and then H of 8 and I'll draw a line here so we don't make it too messy H of 8 we already know is to zero so our function at this endpoint is higher than the value of our function at this endpoint so we do have a positive average rate of change so I would pick that choice right over there and you could see it visually H of six when X is equal to six our value our function is negative nine and when X is equal to eight our value of our function is zero and so the line that connects those two points definitely has a positive slope so we have a positive average rate of change over that interval now if we were just doing this on our own we'd be done but we could just check this one right over here if we compare H of zero we already know is zero and H of six we already know is equal to negative nine so this is a negative average rate of change because at the higher end point right over here we have a lower value of our function so we'd rule this out and you see it right over here if you go from x equals zero where the function is 2x equals six where the function is it looks something it looks something like that clearly that line has a negative slope so we have a negative average rate of change